Question

According to the College Board, SAT mathematics scores from the 2015 school year for high school students in the United States were normally distributed with a mean of 511 and a standard deviation of 120. Determine the probability that a randomly chosen high school student who took the SAT in 2015 will have a mathematics SAT score more than 700 points. Give your answer as a percentage rounded to one decimal place.

Answer 1

5.8%

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 511, \sigma = 120`

Determine the probability that a randomly chosen high school student who took the SAT in 2015 will have a mathematics SAT score more than 700 points.

This is 1 subtracted by the pvalue of Z when X = 700.

`Z = (X - \mu)/(\sigma)`

`Z = (700 - 511)/(120)`

`Z = 1.575`

`Z = 1.575` has a pvalue of 0.942

1 - 0.942 = 0.058

5.8% probability that a randomly chosen high school student who took the SAT in 2015 will have a mathematics SAT score more than 700 points.